Homework Set (2)
This homework set is due on Wednesday (22nd of Dhil-Hijjah) at 10.00 p.m. If you need help, please e-mail (or call) me. In all homeworks, please solve fully and clearly, state assumptions, circle final answers, and comment wisely (when applicable). You are free to discuss with your colleagues, but makes sure your solution is truly yours.
In class, we dealt with the model where we consider two identical harmonic oscillators 1 and 2 separated by R [see textbook, page 53].
a- Derive the first two non-zero terms for the interaction between the two oscillators, and show that for small distortions, equation 3 (of chapter 3) is provides a good approximation.
b- Continue with the coupled oscillators problem (of part-a, above) by diagonalizing the matrix, finding the two modes, and their frequencies.
c- Show, in detail, that the total energy of the coupled system is lower than the independent oscillators.
a- Through a few simple numerical calculations, show that for inert element crystals, it is not important to consider interactions that are not 'close' neighbors. [please use the 6-12 (i.e. Lennard-Jones) potential]
b- Find the equilibrium position of an inert element crystal [assuming the 6-12 potential].
In no more that 8 typed lines each, describe [in your own words] the following:
a- covalent crystals
c- atomic radii
e-mail me (firstname.lastname@example.org) two web-site URLs that clearly demonstrate the different bonding types in materials.